Software

Informal

In the study of complex physical and biological systems represented
by multivariate stochastic processes, an issue of great relevance is
the description of the system dynamics spanning multiple
temporal scales. While methods to assess the dynamic complexity of
individual processes at different time scales are well-established,
multiscale analysis of directed interactions has never been formalized
theoretically, and empirical evaluations are complicated by practical
issues such as filtering and downsampling.

We extend the very popular measure of Granger causality (GC), a
prominent tool for assessing directed lagged interactions between joint
processes, to quantify information transfer across multiple time
scales. We show that the multiscale processing of a vector
autoregressive (AR) process introduces a moving average (MA) component,
and describe how to represent the resulting ARMA process using state
space (SS) models and to combine the SS model parameters for computing
exact GC values at arbitrarily large time scales. We exploit the
theoretical formulation to identify peculiar features of multiscale GC
in basic AR processes, and demonstrate with numerical simulations the
much larger estimation accuracy of the SS approach compared with pure
AR modeling of filtered and downsampled data. The improved computational
reliability is exploited to disclose meaningful multiscale patterns of
information transfer between global temperature and carbon dioxide
concentration time series, both in paleoclimate and in recent years.

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The code is provided free of charge. It is neither exhaustively tested
nor particularly well documented. The authors accept no liability for
its use. Use, modification and redistribution of the code is allowed in
any way users see fit. Authors ask only that authorship is acknowledged
and ref. [1] is cited upon utilization of the code in integral or
partial form.

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WARRANTIES AND LIMITATION OF LIABILITY The code is supplied as is and
all use is at your own risk. The authors disclaim all warranties of any
kind, either express or implied, as to the softwares, including, but
not limited to, implied warranties of fitness for a particular purpose,
merchantability or non - infringement of proprietary rights. Neither
this agreement nor any documentation furnished under it is intended to
express or imply any warranty that the operation of the software will
be error - free. Under no circumstances shall the authors of the
softwares provided here be liable to any user for direct, indirect,
incidental, consequential, special, or exemplary damages, arising from
the software, or user' s use or misuse of the softwares. Such
limitation of liability shall apply whether the damages arise from the
use or misuse of the data provided or errors of the software.

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